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More coin flips) becomes available. The coin will actually be fair, but we won’t learn this until the trials are carried out. At the start we have no prior belief on the fairness of the coin, that is, we can say that any level of fairness is equally likely. 5. We will use a uniform probability distribution as a means of characterising our prior belief that we are unsure about the fairness. This states that we consider each level of fairness (or each value of θ) to be equally likely. We are going to use a Bayesian updating procedure to go from our prior beliefs to posterior beliefs as we observe new coin flips.

In this chapter we will use PyMC3 to carry out a simple example of inferring a binomial proportion. This is sufficient to express the main ideas of MCMC without getting bogged down in implementation specifics. In later chapters we will explore more features of PyMC3 by carrying out inference on more sophisticated models. 5 Inferring a Binomial Proportion with Markov Chain Monte Carlo If you recall from the previous chapter on inferring a binomial proportion using conjugate priors our goal was to estimate the fairness of a coin, by carrying out a sequence of coin flips.

We will use this formula when we come to determine our posterior belief distribution later in the chapter. 5 Quantifying our Prior Beliefs An extremely important step in the Bayesian approach is to determine our prior beliefs and then find a means of quantifying them. In the Bayesian approach we need to determine our prior beliefs on parameters and then find a probability distribution that quantifies these beliefs. In this instance we are interested in our prior beliefs on the fairness of the coin.